Question

The area of the triangle formed by $(0,0),(a^{x^2},0),(0,a^{6x})$ is $\frac{1}{2a^5}sq$ unit then $x$ is equal to

• A. $1$ or $5$
• B. $-1$ or $5$
• C. $1$ or $-5$
• D. $-1$ or $-5$

Answer 1

The correct option is D $-1$ or $-5$

we have,

$A=\frac{1}{2}|x_1(y_2-y_3)+x_2(y_3-y_1)+x_3(y_1-y_2)|$

$\frac{1}{2a^5}=\frac{1}{2}|0+a^{x^2}(a^{6x}-0)+0|$

$\frac{1}{a^5}=a^{x^2}\cdot a^{6x}=a^{x^2+6x}$

$a^{-5}=a^{x^2+6x}$

$-5=x^2+6x$

$x^2+6x+5=0$

$(x+1)(x+5)=0$

$\therefore x=-1,-5$